Using the matrix product formalism, we introduce a two parameter family of exactly solvable $xyz$ spin 1/2 Heisenberg chains in magnetic field (with nearest neighbor interactions) and calculate the ground state and correlation functions in compact form. The ground state has a very interesting property: all the pairs of spins are equally entangled with each other. Therefore it is possible to engineer long-range entanglement in experimentally realizable spin systems on the one hand and study more closely quantum phase transition in such systems on the other.
The features of the concurrences of the nearest-neighbor and the next-nearest-neighbor sites for one-dimensional Heisenberg model with the next-nearest-neighbor interaction are studied both at the ground state and finite temperatures respectively. Both concurrences are found to exhibit different behaviors at the ground state, which is clarified from the point of view of the correlation function. The threshold temperature with respective to different number of sites and the thermal concurrences of the system up to 12 sites are studied numerically.
Various examples of exactly solvable `discrete' quantum mechanics are explored explicitly with emphasis on shape invariance, Heisenberg operator solutions, annihilation-creation operators, the dynamical symmetry algebras and coherent states. The eigenfunctions are the (q-)Askey-scheme of hypergeometric orthogonal polynomials satisfying difference equation versions of the Schrödinger equation. Various reductions (restrictions) of the symmetry algebra of the Askey-Wilson system are explored in detail.
In this article, we study the high order term of the fidelity of the Heisenberg chain with next-nearest-neighbor interaction and analyze its connection with quantum phase transition of Beresinskii-Kosterlitz-Thouless type happened in the system. We calculate the fidelity susceptibility of the system and find that although the phase transition point can't be well characterized by the fidelity susceptibility, it can be effectively picked out by the higher order of the ground-state fidelity for finite-size systems.
The evolution of entanglement in a 3-spin chain with nearest-neighbor Heisenberg-XY interactions for different initial states is investigated here. In an NMR experimental implementation, we generate multipartite entangled states starting from initial separable pseudo-pure states by simulating nearest-neighbor XY interactions in a 3-spin linear chain of nuclear spin qubits. For simulating XY interactions, we follow algebraic method of Zhang et al. [Phys. Rev. A 72, 012331 (2005)]. Bell state between end qubits has been generated by using only the unitary evolution of the XY Hamiltonian. For gen…
We propose a scheme to simulate one-dimensional XXZ-type Heisenberg spin models with competing interactions between nearest-neighbors (NNs) and next-NNs in photon-coupled micro-cavities. Our scheme, for the first time, exploits the rich resources and flexible controls available in such a system to realize arbitrarily adjustable ratios between the effective NN and next-NN coupling strengths. Such powerful capability allows us to simulate frustration phenomena and disorder behaviors in 1-d systems arising from next-NN interactions, a large class of problems of great importance in condensed matte…
We consider quantum systems consisting of a linear chain of n harmonic oscillators coupled by a nearest neighbour interaction of the form $-q_r q_{r+1}$ ($q_r$ refers to the position of the $r$th oscillator). In principle, such systems are always numerically solvable and involve the eigenvalues of the interaction matrix. In this paper, we investigate when such a system is analytically solvable, i.e. when the eigenvalues and eigenvectors of the interaction matrix have analytically closed expressions. This is the case when the interaction matrix coincides with the Jacobi matrix of a system of di…
By using the concept of negativity, we investigate entanglement in (1/2,1) mixed-spin Heisenberg systems. We obtain the analytical results of entanglement in small isotropic Heisenberg clusters with only nearest-neighbor (NN) interactions up to four spins and in the four-spin Heisenberg model with both NN and next-nearest-neighbor (NNN) interactions. For more spins, we numerically study effects of temperature, magnetic fields, and NNN interactions on entanglement. We study in detail the threshold value of the temperature, after which the negativity vanishes.
Next-nearest-neighbor Heisenberg chain plays important roles in solid state physics, such as predicting exotic electric properties of two-dimensional materials or magnetic properties of organic compounds. Direct experimental studies of the many-body electron systems or spin systems associating to these materials are challenging tasks, while optical simulation provides an effective and economical way for immediate observation. Comparing with bulk optics, integrated optics are more of fascinating for steady, large scale and long-time evolution simulations. Photonic crystal is an artificial micro…
Using an analytically solvable model, we show that a qubit array-based detector allows to achieve the fundamental Heisenberg limit in detecting single photons. In case of superconducting qubits, this opens new opportunities for quantum sensing and communications in the important microwave range.
Spin ensembles coupled to optical cavities provide a powerful platform for engineering synthetic quantum matter. Recently, we demonstrated that cavity mediated infinite range interactions can induce fast scrambling in a Heisenberg $XXZ$ spin chain (Phys. Rev. Research {\bf 2}, 043399 (2020)). In this work, we analyze the kaleidoscope of quantum phases that emerge in this system from the interplay of these interactions. Employing both analytical spin-wave theory as well as numerical DMRG calculations, we find that there is a large parameter regime where the continuous $U(1)$ symmetry of this mo…
We study entanglement in dimerized Heisenberg systems. In particular, we give exact results of ground-state pairwise entanglement for the four-qubit model by identifying a Z_2 symmetry. Although the entanglements cannot identify the critical point of the system, the mean entanglement of nearest-neighbor qubits really does, namely, it reaches a maximum at the critical point.
We elucidate the finite temperature entanglement properties of $N=9$ qubits Heisenberg $XX$ and $XXZ$ models under the presence of a polarized magnetic field in $xz$ plane by means of concurrence concept. We perform a systematic analysis for a wide range of the system parameters. Our results suggest that the global phase regions which separate the entangled and non-entangled regions sensitively depend upon the spin-spin interaction term of the $z-$ component of two neighboring spins $J_{z}/J_{x}$, temperature as well as polarized magnetic field components. Thereby, we think that polarized magn…
Recently Levy has shown that quantum computation can be performed using an ABAB.. chain of spin-1/2 systems with nearest-neighbor Heisenberg interactions. Levy notes that all necessary elementary computational `gates' can be achieved by using spin-resonance techniques involving modulating the spin-spin interaction strength at high frequency. Here we note that, as an alternative to that approach, it is possible to perform the elementary gates with simple, non-oscillatory pulses.
For a large class of time-dependent non-Hermitain Hamiltonians expressed in terms linear and bilinear combinations of the generators for an Euclidean Lie-algebra respecting different types of PT-symmetries, we find explicit solutions to the time-dependent Dyson equation. A specific Hermitian model with explicit time-dependence is analyzed further and shown to be quasi-exactly solvable. Technically we constructed the Lewis-Riesenfeld invariants making useof the metric picture, which is an equivalent alternative to the Schrödinger, Heisenberg and interaction picture containing the time-dependenc…
Hamiltonian-Generated Dynamics and Quantum Resource Allocation in a 4-Qubit Isotropic Heisenberg Spin Ring with Adjustable Next-Nearest-Neighbor Interactions
A new approach in solution of simple quantum mechanical problems in deformed space with minimal length is presented. We propose the generalization of Schroëdinger equation in momentum representation on the case of deformed Heisenberg algebra with minimal length. Assuming that the kernel of potential energy operator do not change in the case of deformation, we obtain exact solution of eigenproblem of a particle in delta potential as well as double delta potential. Particle in Coulomb like potential is revisited and the problem of inversibility and hermicity of inverse coordinate operator is sol…
In this paper, we convert the lattice configurations into networks with different modes of links and consider models on networks with arbitrary numbers of interacting particle-pairs. We solve the Heisenberg model by revealing the relation between the Casimir operator of the unitary group and the conjugacy-class operator of the permutation group. We generalize the Heisenberg model by this relation and give a series of exactly solvable models. Moreover, by numerically calculating the eigenvalue of Heisenberg models and random walks on network with different numbers of links, we show that a syste…
Localization properties of non-interacting quantum particles in one-dimensional incommensurate lattices are investigated with an exponential short-range hopping that is beyond the minimal nearest-neighbor tight-binding model. Energy dependent mobility edges are analytically predicted in this model and verified with numerical calculations. The results are then mapped to the continuum Schrodinger equation, and an approximate analytical expression for the localization phase diagram and the energy dependent mobility edges in the ground band obtained.
A comprehensive review of exactly solvable quantum mechanics is presented with the emphasis of the recently discovered multi-indexed orthogonal polynomials. The main subjects to be discussed are the factorised Hamiltonians, the general structure of the solution spaces of the Schroedinger equation (Crum's theorem and its modifications), the shape invariance, the exact solvability in the Schroedinger picture as well as in the Heisenberg picture, the creation/annihilation operators and the dynamical symmetry algebras, coherent states, various deformation schemes (multiple Darboux transformations)…
In an innovative inverse-problem construction the measured, experimental energies $E_1$, $E_2$, ...$E_N$ of a quantum bound-state system are assumed fitted by an N-plet of zeros of a classical orthogonal polynomial $f_N(E)$. We reconstruct the underlying Hamiltonian $H$ (in the most elementary nearest-neighbor-interaction form) and the underlying Hilbert space ${\cal H}$ of states (the rich menu of non-equivalent inner products is offered). The Gegenbauer's ultraspherical polynomials $f_n(x)=C_n^α(x)$ are chosen for the detailed illustration of technicalities.
Brief introduction to the discrete quantum mechanics is given together with the main results on various exactly solvable systems. Namely, the intertwining relations, shape invariance, Heisenberg operator solutions, annihilation/creation operators, dynamical symmetry algebras including the $q$-oscillator algebra and the Askey-Wilson algebra. A simple recipe to construct exactly and quasi-exactly solvable Hamiltonians in one-dimensional `discrete' quantum mechanics is presented. It reproduces all the known ones whose eigenfunctions consist of the Askey scheme of hypergeometric orthogonal polynom…
